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2   SPECTRUM OF AN ELEMENT OF A NORMED ALGEBRA      317

(e)    Let / be an analytic function on a neighborhood of Sp(*), and g an analytic
function on a neighborhood of Sp(/(*)). Show that, by restricting the open set on
which/is defined, if necessary, the function h = g o/is defined, and that h(x) = g(f(x)).

(f)    In particular, if Sp(x) is contained in the disk || < 1, then for each a e Rthe
series

converges in A to an element y which commutes with x, and SpQO is the image of
Sp(.x) under the mapping H->(! +) (with the determination (1 +)=! when
 = 0). In particular, if a = 1/m where m is an integer > 0, then ym = 1 -f x, and we
write

12.   With the same hypotheses as in Problem 11, let (Kj)1<y<n be a partition of Sp(#)
into compact sets. For each;, let U/ be a neighborhood of K,, such that the U/ are

n

pairwise disjoint. Let// be the analytic function on (J Ufc which takes the value 1 on

*=i

U/ and 0 on Ufc for k^j.  Show that ej = //Jc) is idempotent in A, that e}ek = 0

n

whenever A: ^=y, and that V e/  e .
/-i

(b)   Let A be an isolated point of Sp(*), and let

be the Laurent expansion of (&  x)'1 ina neighborhood of A. If eA is the idempotent
corresponding to {A} in the partition of Sp(x) consisting of {A} and Sp(jt)  {A}, then
e^Xe - x)k = (-l)^.^! for all k ;> 0.

(c)    Let B be the subalgebra of A generated by e and x. In order that Sp(jc) should
consist of only a finite number of points, which are poles of (e  x)"1, it is necessary

r

and sufficient that B should be finite-dimensional. (If (e  x)~l = V F/()fl/ where

j=i

the F, are scalar rational functions, then all powers of x are linear combinations of
the aj . Conversely, if B is finite-dimensional, there exists a polynomial/, not identically
zero, such that/(x) = 0. Then Sp(x) is contained in the set of zeros of/; if A is a zero
of/, of order/?, show also that (Xe  x)pe^ = 0.)

(d)    Let U be an open neighborhood of Sp(x), having only finitely many connected
components U, , and let / be an analytic function on U. Show that f(x) = 0 if and
only if the following conditions are satisfied: (1) for each index j such that Sp(jc) n Uj
is either infinite or else contains an essential singularity of (e  x)"1, we have
f\Uj  0; (2) every pole of order p of (&  x)""1 is a zero of/, of order ^p. (For the
necessity of (2), argue as in (c).)

13.    Let A be a Banach algebra with unit element, and U an open set in C. Show that
the set D of elements x e A such that SpA(x) c U is open in A. If / is analytic on U,
then the mapping x\ *f(x) of O into A is indefinitely differentiate.

14.    Let A be a Banach algebra with unit element denoted by 1 .

(a)   For each pair jc, y of elements of A and each  e C, show that p(e^xye"tix)  p(y).if and only if /() ^ 0 on Sp(#).n particular, the image of C under the